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The Phenomenon of Quenching for a Reaction-Diffusion System with Non-Linear Boundary Conditions


Affiliations
1 Department of Mathematics and Informatics, Universite Nangui Abrogoua, UFR-SFA, 02 BP 801 Abidjan, Côte d'Ivoire
2 Department of Mathematics and Informatics, Universite Peleforo Gon Coulibaly de Korhogo, UFR-Sciences Biologiques, BP 1328 Korhogo, Côte d'Ivoire
3 Department of Mathematics and Informatics, Universite Alassane Ouattara de Bouake, UFR-SED, 01 BP V 18 Bouake 01, Côte d'Ivoire
     

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We study the quenching behavior of the solution of a semi- linear reaction-diffusion system with nonlinear boundary conditions. We prove that the solution quenches in finite time and its quenching time goes to the one of the solution of the differential system. We also obtain lower and upper bounds for quenching time of the solution.

Keywords

Quenching, reaction-diffusion system, finite difference, numerical quenching time, nonlinear boundary condition, maximum principles
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  • T. K. Boni, Extinction for discretizations of some semilinear parabolic equations, C.R.A.S, Serie I, 333 (2001), 795-800.

  • T. K. Boni, H. Nachid and F. K. N'gohisse, Quenching time for a nonlocal diffusion problem with large initial data, Bull. Iran. Math. Soc., 35(2) (2009), 191-208.

  • C. Y. Chan, New results in quenching. In Proceedings of the First World Congress of Nonlinear Analysts., Walter de Gruyter, New York, USA, (1996), 427-434.

  • C. Y. Chan, Recent advances in quenching phenomena, Proc. Dynam. Sys. Appl., 2, 1996, 107-113.

  • C. Y. Chan and Ozalp N. Beyond, quenching for singular reaction-diffusion mixed boundary-value problems., In Advances in Nonlinear Dynamics, Ed. S. Sivasundaram and A. A. Martynyuk, Gordon and Breach, Amsterdam, (2001), 217-227.

  • C. W. Chang, Y. H. Hsu and H. T. Liu, Quenching behavior of parabolic problems with localized reaction term., IMathematics and Statistics, 2, (2014), 48-53.

  • C. Y. Chan and S. I. Yuen, Parabolic problems with nonlinear absorptions and releases at the boundaries, Appl. Math. Comput, 121 (2001), 203-209.

  • A. Friedman and A. A. Lacey, The blow-up time for solutions of nonlinear heat equations with small diusion, SIAM J. Math. Anal., 18 (1987), 711-721.

  • S. C. Fu and J. S. Guo, Blow up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions. J. Math. Anal. Appl., 276 (2002) 458-475.

  • G. Gui and X. Wang, Life span of solutions of the Cauchy problem for a nonlinear heat equation, J. Di. Equat., 115 (1995), 162-172.

  • H. Kawarada, On solutions of initial-boundary problem for ut = uxx+1=(1-u). Publ. Res. Inst. Math. Sci., 10 (1975), 729-736.

  • C. M. Kirk and C. A. Roberts, A review of quenching results in the context of nonlinear volterra equations. Dynam. Cont. Dis. Ser. A, 10 (2003), 343-356.

  • D. Nabongo and T. K. Boni, Quenching time of solutions for some nonlinear parabolic equations, An. St. Univ. Ovidus Constanta, 16(1) (2008), 91-106.

  • A. De Pablo, F. Quiros and J. D. Rossi, Nonsimultaneous Quenching, Appl. Math. Letters, 15 (2002) 265-269.

  • B. Selcuk and N. Ozalp, The quenching behavior of a semilinear heat equation with a singular boundary outflux. Q. Appl. Math., 72 (2014), 747-752.


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  • The Phenomenon of Quenching for a Reaction-Diffusion System with Non-Linear Boundary Conditions

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Authors

Halima Nachid
Department of Mathematics and Informatics, Universite Nangui Abrogoua, UFR-SFA, 02 BP 801 Abidjan, Côte d'Ivoire
F. N'Gohisse
Department of Mathematics and Informatics, Universite Peleforo Gon Coulibaly de Korhogo, UFR-Sciences Biologiques, BP 1328 Korhogo, Côte d'Ivoire
N'Guessan Koffi
Department of Mathematics and Informatics, Universite Alassane Ouattara de Bouake, UFR-SED, 01 BP V 18 Bouake 01, Côte d'Ivoire

Abstract


We study the quenching behavior of the solution of a semi- linear reaction-diffusion system with nonlinear boundary conditions. We prove that the solution quenches in finite time and its quenching time goes to the one of the solution of the differential system. We also obtain lower and upper bounds for quenching time of the solution.

Keywords


Quenching, reaction-diffusion system, finite difference, numerical quenching time, nonlinear boundary condition, maximum principles

References





DOI: https://doi.org/10.18311/jims%2F2021%2F26056